Fixed point (mathematics)

A function with three fixed points

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, c is a fixed point of the function f if f(c) = c. This means f(f(...f(c)...)) = fⁿ(c) = c, an important terminating consideration when recursively computing f. A set of fixed points is sometimes called a fixed set.

For example, if f is defined on the real numbers by

${\displaystyle f(x)=x^{2}-3x+4,}$

then 2 is a fixed point of f, because f(2) = 2.

Not all functions have fixed points: for example, f(x) = x + 1, has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point x means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.

Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point.^[1][2]

In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.

Attracting fixed points[]

The fixed point iteration x_n+1 = cos x_n with initial value x₁ = −1.

An attracting fixed point of a function f is a fixed point x₀ of f such that for any value of x in the domain that is close enough to x₀, the iterated function sequence

${\displaystyle x,\ f(x),\ f(f(x)),\ f(f(f(x))),\dots }$

converges to x₀. An expression of prerequisites and proof of the existence of such a solution is given by the Banach fixed-point theorem.

The natural cosine function ("natural" means in radians, not degrees or other units) has exactly one fixed point, which is attracting. In this case, "close enough" is not a stringent criterion at all—to demonstrate this, start with any real number and repeatedly press the cos key on a calculator (checking first that the calculator is in "radians" mode). It eventually converges to about 0.739085133, which is a fixed point. That is where the graph of the cosine function intersects the line ${\displaystyle y=x}$.^[3]

Not all fixed points are attracting. For example, x = 0 is a fixed point of the function f(x) = 2x, but iteration of this function for any value other than zero rapidly diverges. However, if the function f is continuously differentiable in an open neighbourhood of a fixed point x₀, and ${\displaystyle |f\,'(x_{0})|<1}$, attraction is guaranteed.

Attracting fixed points are a special case of a wider mathematical concept of attractors.

An attracting fixed point is said to be a stable fixed point if it is also Lyapunov stable.

A fixed point is said to be a neutrally stable fixed point if it is Lyapunov stable but not attracting. The center of a linear homogeneous differential equation of the second order is an example of a neutrally stable fixed point.

Multiple attracting points can be collected in an attracting fixed set.

Applications[]

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In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. Some examples follow.

• In economics, a Nash equilibrium of a game is a fixed point of the game's best response correspondence. John Nash exploited the Kakutani fixed-point theorem for his seminal paper that won him the Nobel prize in economics.

• In physics, more precisely in the theory of phase transitions, linearisation near an unstable fixed point has led to Wilson's Nobel prize-winning work inventing the renormalization group, and to the mathematical explanation of the term "critical phenomenon."^[4][5]

• Programming language compilers use fixed point computations for program analysis, for example in data-flow analysis, which is often required for code optimization. They are also the core concept used by the generic program analysis method abstract interpretation.^[6]

• In type theory, the fixed-point combinator allows definition of recursive functions in the untyped lambda calculus.

• The vector of PageRank values of all web pages is the fixed point of a linear transformation derived from the World Wide Web's link structure.

• The stationary distribution of a Markov chain is the fixed point of the one step transition probability function.

• Logician Saul Kripke makes use of fixed points in his influential theory of truth. He shows how one can generate a partially defined truth predicate (one that remains undefined for problematic sentences like "This sentence is not true"), by recursively defining "truth" starting from the segment of a language that contains no occurrences of the word, and continuing until the process ceases to yield any newly well-defined sentences. (This takes a countable infinity of steps.) That is, for a language L, let L′ (read "L-prime") be the language generated by adding to L, for each sentence S in L, the sentence "S is true." A fixed point is reached when L′ is L; at this point sentences like "This sentence is not true" remain undefined, so, according to Kripke, the theory is suitable for a natural language that contains its own truth predicate.

Topological fixed point property[]

A topological space ${\displaystyle X}$ is said to have the fixed point property (FPP) if for any continuous function

${\displaystyle f\colon X\to X}$

there exists ${\displaystyle x\in X}$ such that ${\displaystyle f(x)=x}$.

The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The FPP is also preserved by any retraction.

According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.^[7]

Generalization to partial orders: prefixpoint and postfixpoint[]

The notion and terminology is generalized to a partial order. Let ≤ be a partial order over a set X and let f: X → X be a function over X. Then a prefixpoint (also spelled pre-fixpoint) of f is any p such that p ≤ f(p). Analogously, a postfixpoint (or post-fixpoint) of f is any p such that f(p) ≤ p.^[8] One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixpoint that coincides with its least postfixpoint (and similarly its greatest fixpoint coincides with its greatest prefixpoint). Prefixpoints and postfixpoints have applications in theoretical computer science.^[9]

See also[]

Notes[]

External links[]

• An Elegant Solution for Drawing a Fixed Point